For a fixed mmm, the limit limk→∞⁡Sksubscriptnormal-→ksubscriptSk\lim_{{k\to\infty}}S_{k} apparently
exists iff the limit
limn→∞⁡Sm(n)subscriptnormal-→nsuperscriptsubscriptSmn\lim_{{n\to\infty}}S_{m}^{{(n)}} exists. Thus we can write the
Theorem. The series (1) is convergent if and only if
each remainder term series (2) is convergent.

Mathematics Subject Classification

Comments

Why the Cyrillic letters in the References of http://planetmath.org/remaindertermseries don’t work? In an older entryhttp://planetmath.org/divisorasfactorofprincipaldivisor they work well. Both have similar preambles.